The main
objective of the current article is to verify in Theorem REF  that this condition is also sufficient for joint ergodicity, assuming that the sequences \(a_1,\ldots ,a_\ell \)  satisfy another necessary condition regarding seminorm estimates that is defined in the Section REF .
The necessary conditions we give are flexible enough to allow us to recover painlessly all the known results of jointly ergodic sequences mentioned above and enable us to
prove some new ones. For example, in Theorem REF  we make progress towards a conjecture in
[1]} regarding joint ergodicity of Hardy field sequences,
improving upon the best known results in [2]}, [1]}, and in Theorem REF  we verify a conjecture from [2]}. A special case of this last result asserts that a collection of sequences of the form
\([\sum _{i=1}^k\alpha _in^{b_i}]\) , where \(\alpha _1,\ldots , \alpha _k\in {\mathbb {Q}}\) , \(b_1,\ldots , b_k\in (0,+\infty )\)  are positive,
is jointly ergodic for all totally ergodic systems if and if it is linearly independent;
this was previously known when all the exponents are integers [5]} or none of the exponents is an integer [2]}.
